The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

Nonfiction, Science & Nature, Mathematics, Mathematical Analysis, Statistics
Cover of the book The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise by Arnaud Debussche, Michael Högele, Peter Imkeller, Springer International Publishing
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Author: Arnaud Debussche, Michael Högele, Peter Imkeller ISBN: 9783319008288
Publisher: Springer International Publishing Publication: October 1, 2013
Imprint: Springer Language: English
Author: Arnaud Debussche, Michael Högele, Peter Imkeller
ISBN: 9783319008288
Publisher: Springer International Publishing
Publication: October 1, 2013
Imprint: Springer
Language: English

This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.

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This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.

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